About €40-45 on the meter, taking an official taxi from the rank outside the Arrivals area, where waiting taxis have to be registered in Flanders (the airport being outside the limit of the city and Region of Brussels). Can be cheaper if you book a private transfer at a flat rate with a Brussels taxi company, e.g. www.autolux.be
It isn't possible for any driver (let alone car!) to enter the area of the airport where bags are delivered, as this is before Customs and only airport staff can operate "airside". You can either:
- reserve an airport porter, see brusselsairport.be/en/… or
- pick up your own bags from the carrousel, put them on a free baggage cart, and push this through Customs to the exit "landside", where, if you have booked a transfer service, you would be met by the driver with a sign. He would then escort you to wherever the car is parked (which cannot be directly outside as this is for metered taxis) and I imagine he would push the cart for you and certainly load the bags into the car.
Incidentally, it seems this hotel at Place Stéphanie isn't the Conrad any more but the Steigenberger Grand. However, taxi drivers will no doubt still know it as the Conrad.